Question

If the sum of n terms of an A.P. is $2{n^2} + 5n$ , then the $n^{th}$ term will be
A. 4n+3
B. 4n+5
C. 4n+6
D. 4n+7

Answer 1

Given is the sum of n terms of the A.P. we know that in order to find the nth term we need the value of the first term and the common difference also. For that we will use the given formula of sum and find the first term and the sum of the first two terms. then we can find the common difference also. After that, using the formula of nth term, we will find the correct answer.
Formula used:
nth term of an A.P. is given by ${T_n} = a + \left( {n - 1} \right)d$
where a is the first term and d is the common difference.

Complete step by step answer:
Given that $2{n^2} + 5n$ is the sum of n terms of an A.P.
Foe n=1 we can write, $2 \times 1 + 5 = 2 + 5 = 7$
Thus this is the first term a=7.
Now we will find the sum of the first two terms putting n=2 in the formula itself.
${S_2} = 2 \times {2^2} + 5 \times 2 = 8 + 10 = 18$
Now the difference between the sum and the first term will be the common difference.
$18 - 7 = 11$
Thus d=4 is the common difference. Since the A.P. will be 7,11,15,…
Now we have first term a and common difference d. putting these values in the formula above we get the nth term as,

$${T_n} = a + \left( {n - 1} \right)4 \\\ {T_n} = 7 + \left( {n - 1} \right)4 \\\ {T_n} = 7 + 4n - 4 \\\ {T_n} = 4n + 3 \\\$$

This is the formula or the nth term.
So, the correct answer is “Option A”.

Note: Note that, the difference we have found here is between the first term and the sum of the first two terms and is the same as the common difference. But this is not applicable for the difference between sum of first three terms and the first term. Remember this!
Don’t try to form any equation or any substitution. This is the best approach to solve.
Remember the formula for the sum of n terms and to find nth term is totally different.